Omni-Lie Algebras

Details Omni-Lie Algebras Omni-Lie Algebras

1999-12-22

arxiv.org/abs/math/9912190v2

Mathematics

Representation Theory

8 pages, minor corrections and announcement of further results

Scientific paper

We show that the space R^n x gl(n,R) with a certain antisymmetric bracket operation contains all n-dimensional Lie algebras. The bracket does not satisfy the Jacobi identity, but it does satisfy it for subalgebras which are isotropic under a certain symmetric bilinear form with values in R^n. We ask what the corresponding "group-like" object should be. The bracket may be obtained by linearizing at a point the bracket on TM + T*M introduced by T. Courant for the definition of Dirac structures, a notion which encompasses Poisson structures, closed 2-forms, and foliations.

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